1. Interpolating the INO magnetic field map
Sanjeev Kumar and Md. Naimmuddin

2. Locating the cell
To calculate the field at Point P (x, y), we need the field on the four corners of the cell ACBD containing the point P
Here, A(x1,y1), B(x2,y2), C(x1,y2),D(x2,y1)
This can be calculated from the coordinates of the point P and the size of the cell

3. Interpolation of INO magnetic field map

4. Interpolation of INO magnetic field map
Locate the cell ACBD containing the point P

5. How the interpolation works

6. How the interpolation works
Fields at A and D are used to get the field at F

7. How the interpolation works
Fields at C and B are used to
get the field at G
Fields at A and D are used to get the field at F

8. How the interpolation works
Fields at C and B are used to
get the field at G
Finally, fields at F and G are used to get the field at P
Fields at A and D are used to get the field at F

9. Bilinear spline interpolation
Bilinear spline interpolation means a spline interpolation of order one (linear) in two dimensions (bi).
The interpolated field at point P is given by
f ( x, y) = a + b x + c y + d x y
Here, the coefficients a, b, c and d are some weighted averages of the fields at the four corners of the cell containing P

10. Why spline?
The coefficients a, b, c and d are defined for a cell and vary from cell to cell.
The field at a point on the boundary of two cells can be obtained from two different sets of coefficients corresponding to the two cells.
These two values match : this is the condition from which we determine the expressions for the coefficients in terms of the fields on the corners.

11. Why spline? This is the general principle behind spline interpolation:
If the field values should match at the boundaries it is first order spline interpolation.
If the field values and their derivatives should match at the boundaries it is second order spline interpolation.
If the field values and their first and second order derivatives should match at the boundaries, it is third order spline interpolation.

12. Why linear?
Linear interpolation is used to get the field at F from the field values at A and D, at G from B and C and at P from F and G.
Hence, the interpolated field at P depends only on the field values on the corners of the cell containing it and not on the field in the neighboring cells.
Interpolation is linear in this sense.

13. Not a linear function!
However, the interpolation function is not really linear: it contains a second order term (d x y )
A linear function of the form
z( x, y) = a + b x + c y
is a plane which cannot always pass through the four points (x1,y1,z11), (x1,y2,z12), (x2,y1,z21), (x2,y2,z22) in three dimensions
Hence, we need the term (d x y )

14. Higher order splines
As said earlier, if the field values and their first and second order derivatives should match at the boundaries, it is third order spline interpolation.
A third order spline interpolation in two dimensions is called bicubic spline interpolation.
The interpolated field in the bicubic spline will depend on the field values in all the neighboring cells as well.

15. Bilinear interpolation function

16. Program for the bilinear interpolation function

17. Absolute errors in Bx and By

18. Percentage errors in Bx and By

19. Errors normalized to B

20. Errors in bilinear interpolation
Errors in bilinear interpolations of By are less than 0.5% virtually everywhere. The rms value is as small as 0.1%.
Errors in Bx can be as large as 10% in some regions of the grid. The rms value is 3%.
However, the regions where errors in Bx are large are the regions where Bx is small and By is large. Therefore, when we normalize errors in Bx w.r.t B, the rms value reduces to 0.5%.

21. Bicubic spline interpolation
It has been found that the interpolation errors in Bx can be further reduced if we use bicubic spline interpolation.
A study using Mathematica has been already done.
C++ implementation will be done soon.